arX

iv:a

stro

-ph/

0404

520v

1 2

7 A

pr 2

004

Effect of Strong Quantizing Magnetic Field on the Transport Properties of Dense

Stellar Plasma

Soma Mandala) and Somenath Chakrabartya),b)∗

a)Department of Physics, University of Kalyani, Kalyani 741 235, India and b)Inter-University Centre for Astronomy and

Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India

(February 2, 2008)

PACS:97.60.Jd, 97.60.-s, 75.25.+z

The transport properties of dense stellar electron-proton plasma is studied following an exactrelativistic formalism in presence of strong quantizing magnetic field. The variation of transportcoefficients with magnetic fields are found to be insensitive for the field strengths ≤ 1017G, beyondwhich all of them abruptly go to zero. As a consequence, the electron-proton plasma behaves like asuperfluid insulator in presence of ultra-strong magnetic field.

1. INTRODUCTION

The study of the effect of strong magnetic field on dense stellar plasma is one of the oldest branches of physics. Ithas gotten a new life after the discovery of a few magnetars- the strange stellar objects, with unusually high surfacemagnetic fields [1–4]. These stellar objects are believed to be strongly magnetized young neutron stars. The surfacemagnetic fields are observed to be ∼ 1015G. Then it is quite possible that the field strength at the core region maygo up to 1018G. The exact source of this strong magnetic field is of course yet to be known. These objects are alsosupposed to be the possible sources of anomalous X-ray and soft gamma emissions (AXP and SGR). If the magneticfields are really so strong, in particular at the core region, they must affect significantly most of the important physicalproperties of such stellar objects and the physical processes, e.g., weak and electromagnetic interactions taking placeat the core region. Which means, the presence of strong quantizing magnetic field at the core region should modify,both qualitatively and quantitatively the equation of state of dense neutron star matter, and as a consequence thegross-properties of neutron stars [5–8], e.g., mass-radius relation, moment of inertia, rotational frequency etc. shouldalso change significantly. In the case of compact neutron stars, the phase transition from neutron matter to quarkmatter at the core region, if any, will also be affected by strong quantizing magnetic field. It has been shown that afirst order phase transition initiated by the nucleation of quark matter droplets is absolutely forbidden if the magneticfield ∼ 1015G at the core region [9,10]. However a second order phase transition is allowed, provided the magneticfield strength < 1020G. This is of course too high to achieve at the core region.

The elementary processes, in particular, the weak and the electromagnetic processes taking place at the core regionof a neutron star are strongly affected by such ultra-strong magnetic field [11,12]. Since the cooling of neutron starsare mainly controlled by neutrino/anti-neutrino emissions, the presence of strong quantizing magnetic field shouldaffect the thermal history of strongly magnetized neutron stars. Further, the electrical conductivity of neutron starmatter which directly controls the evolution of neutron star magnetic field will also change significantly.

In another kind of work, the stability of such strongly magnetized rotating objects are studied. It has been observedfrom the detailed general relativistic calculation that there are possibility of some form of geometrical deformationof these objects from their usual spherical shapes [13–15]. In the extreme case such objects may either become blackstrings or black disks. In the non-extreme case, however, it is also possible to detect gravity waves from these deformedrotating objects.

In a recent study on microscopic model of dense neutron star matter, we have observed that if most of the electronsoccupy the zeroth Landau level, with spin anti-parallel to the direction of magnetic field and very few of them are withspin along the direction of magnetic field and Landau quantum number > 0, then either such strongly magnetizedsystem can not exist or such a strong magnetic field is just impossible at the core region of a neutron star [16].

Motivated by the problems as mentioned in preceding two sections, in this paper we shall study the effect of strongquantizing magnetic field on the transport coefficients of dense stellar electron-proton plasma. We shall follow an exactformalism [17] which is applicable for a wide range of magnetic field strengths and obtain the transport coefficientsfrom the relativistic version of Boltzmann kinetic equation by linearizing the distribution function and using relaxationtime approximation. We shall obtain the relaxation time from the rates of standard electromagnetic processes taking

∗E-Mail: somenath@klyuniv.ernet.in

1

place inside electron-proton plasma and make necessary modification in the rate calculation due to the presence ofstrong quantizing magnetic field. We have noticed that the electrical conductivity of the medium becomes extremelysmall in presence of ultra-strong magnetic field (≥ 1017G). The magnetic field at the core region of a magnetar musttherefore decay very rapidly (time scale ∼ a few mins.) and becomes moderate or low enough. As a consequence therewill be in principle no problem on the existence of magnetars (with very low or moderate core magnetic field). Theformalism we have developed to obtain rates of electromagnetic processes or the relaxation time is also applicable toevaluate neutrino emissivity and mean free path in presence of strong quantizing magnetic field.

In presence of strong quantizing magnetic field, since the momentum component in the transverse plane with respectto the external magnetic field gets quantized, whereas the component along the field direction varies continuously(from −∞ to +∞), the momentum space volume element becomes

d3p

(2π)3=

dpxdpydpz

(2π)3=

eBm

4π2

∞∑

ν=0

(2 − δν0)dpz (1)

(we have assumed h = c = kb = 1) where we have chosen the gauge Aµ ≡ (0, 0, xBm, 0), so that the constant magneticfield Bm is along z-direction. We have considered the simplest possible picture of neutron star matter with n− p− eout of thermodynamic equilibrium and the neutrinos are assumed to be non-degenerate. The baryonic componentsare interacting via σ − ω − ρ meson exchange type mean field and the electrons are assumed to be freely movingparticles.

In this article, we shall first calculate the transport coefficients of electron gas. Then it is very easy to obtain thetransport coefficients for proton matter just by replacing mass, chemical potential etc. of electrons by protons andtaking into account the proper modification in presence of σ − ω − ρ meson exchange type mean field [6]. Spinorsolutions for electrons in presence of strong quantizing magnetic fields are then given by

Ψ(↑)(e) =1

√

LyLz

exp(−iε(e)ν t + ipyy + ipzz)

[2ε(e)ν (ε

(e)ν + me)]1/2

×

(ε(e)ν + me)Iν;py

(x)0

pzIν;py(x)

−i(2νeBm)1/2Iν−1;py(x)

(2)

and

Ψ(↓)(e) =1

√

LyLz

exp(−iε(e)ν t + ipyy + ipzz)

[2ε(e)ν (ε

(e)ν + me)]1/2

×

0

(ε(e)ν + me)Iν−1;py

(x)i(2νeBm)1/2Iν;py

(x)−pzIν−1;py

(x)

(3)

where ↑ and ↓ represent up and down spin states respectively,

Iν;py(x) =

(

eBm

π

)1/41√

ν! 2ν/2× exp

[

−1

2eBm

(

x − py

eBm

)2]

Hν

[

√

eBm

(

x − py

eBm

)]

, (4)

with Hν the Hermite polynomial of order ν and

εeν = (p2

z + m2e + 2eνBm)1/2 (5)

the energy eigen value with ν = 0, 1, 2, ....., the Landau quantum numbers. For neutron we consider the usualspinor solutions. Since the temperature of the system is ≪ electron chemical potential, we have not considered thenegative energy spinor solutions, In the energy eigen value p⊥ = (2νeBm)1/2 is the transverse component of electronmomentum. To overcome the serious problem on the mechanical stability and hence the existence of magnetars, wehave studied the variation of transport coefficients, in particular the electrical conductivity of electron gas / protonmatter in presence of strong quantizing magnetic field and try to show whether the electrical conductivity whichis solely responsible for the evolution of neutron star magnetic field, becomes sufficiently small in presence of ultrastrong magnetic field.

The paper is organized in the following manner. In section 2, we have developed the relativistic version of Boltzmannkinetic equation for fermions in presence of strong quantizing magnetic field and obtain the expressions for transportcoefficients following de Groot [18]. We shall use the relaxation time approximation for the collision term. In section 3we shall evaluate the the relaxation time from the rates of standard electromagnetic processes taking place at the coreregion. We shall incorporate the necessary changes in the rate calculation due to the presence of strong quantizingmagnetic field (or the effect of quantized Landau levels). In the last section we shall discuss importance of the results.

2

2. BOLTZMANN EQUATION

The relativistic version of Boltzmann transport equation is given by [18]

pµ∂µf + eFµνpν∂f

∂pµ+ Γµ

νλpνpλ ∂f

∂pµ= C (6)

where the second and the third terms are coming from electromagnetic and gravitational (general relativistic) inter-actions respectively and C is the collision term. Since we have considered the flat space-time geometry and noticedthat there is no contribution of external magnetic field, only the induced electric field arising from local charge non-neutrality, causes an induced current in the system will contribute in the electromagnetic interaction term and thecurvature term is neglected.

To obtain the shear and bulk viscosity coefficients, heat conductivity and electrical conductivity of dense electrongas we make the relaxation time approximation, given by

C = −p0

τ

(

f(x, p) − f0(p))

(7)

where

f(x, p) = f (0)(p) (1 + χ(x, p)) (8)

i.e., the system is assumed to be very close to the local equilibrium configuration. Here τ is the relaxation time andf0 is the (local) equilibrium distribution (Fermi distribution) function, given by.

f (0)(p) =1

exp β(ǫν − µe) + 1(9)

We shall now follow the general technique to obtain the perturbed part χ as a linear sum of driving forces. Followingthe formalism developed in the book by de Groot [18], we express the four derivative as the sum of a space like partand a time like part, given by ∂µ = uµD + ∇µ with uµ the hydrodynamic velocity, D = uµ∂µ is the convectivetime derivative and ∇µ = ∆µν∂ν is the gradient operator, with ∆µν = gµν − uµuν some kind of projection operator,gµν = diag(1,−1,−1,−1) the metric tensor. Then using the eqns.(6)-(9), the decomposition of ∂µ and the equationsof motion, given by

Dn = −n∇µuµ, Duµ =1

nh∇µP, CvDT = −F (T )∇µuµ

we obtain the perturbative part χ, given by

Tp0

τχ = QX

(

1 − f (0))

− (pµuµ − h)pνXqν(

1 − f (0))

+ pµpνX0µν

(

1 − f (0))

+ e

[

pµuµ

h+ 1

]

pνEν(1 − f (0)) (10)

where

X = −∇µuµ (11)

is the driving force for bulk-viscosity,

Xµq = ∇µT − T

nh∇µP (12)

is the driving force for heat conduction,

X0µν = ∇µuν − 1

3∆µν∇σuσ (13)

is the driving force for shear viscosity and Eν is the driving force for electric current (Eν for ν = i = 1, 2, 3 are thecomponents of electric field vector). The quantity Q is given by

3

Q = −1

3∆µνpµpν + (pµuµ)2

F (T )

T(1 − γ) +

(

T 2(1 − γ)F (T )

T

∂

∂T

( µ

T

)

− n∂µ

∂n

)

pµuµ (14)

with F (T ) = P (T )/n(T ), P (T ) and n(T ) are the equilibrium local kinetic pressure and number density respectively.For a non-relativistic Boltzmann gas F (T ) = T , the local temperature of the system and finally h = (ǫ + P )/n, theenthalpy per particle with ǫ the local energy density and γ = Cp/Cv the ratio of specific heats at constant pressureand constant volume respectively.

Now from the definition, the heat flow four current is given by

Iµq =

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0(pνuν − h) pµf(x, p)

= I(0)µq + I(1)µ

q (15)

where the first term is the equilibrium contribution, which is identically zero. Then omitting the symbol (1), we havethe irreversible term

I(1)µq = Iµ

q =eBm

4π2

∑

ν

(2 − δnu0)

∫ +∞

−∞

dpz

p0pµ(pνuν − h)f (0)(p)χ(x, p) (16)

Again using the definition

Iqµ = λµνXqν , (17)

we have the heat conductivity coefficient

λ =1

3∆µνλµν = − 1

3T 2

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p20

τ(pσuσ − h)2f (0)(1 − f (0))∆µνpµpν (18)

Now from the definition, the energy-momentum tensor is given by

T µν =eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0pµpνf(x, p) (19)

Which can also be written as T µν = T (0)µν + T (1)µν , with the equilibrium value

T (0)µν =eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0pµpνf (0)(p) (20)

and the non-equilibrium part

T (1)µν = T µν =eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0pµpνχ(x, p)f (0)(p) (21)

which is a symmetric second rank tensor. Now we consider a model in which a flow of electron gas with cylindricalsymmetry is assumed. Then considering µ = r, ν = z, we have from the definition

T (1)rz = −ηduz

dr(22)

Hence the shear viscosity coefficient is given by

η =eBm

4π2T

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p20

(prpz)2f (0)(1 − f (0))τ (23)

where pr = (2νeBm)1/2, the transverse component of electron momentum. Then it is quite obvious that in presenceof ultra strong magnetic field, for which νmax = 0, the shear viscosity coefficient vanishes. Here νmax is the maximumvalue of quantum number of the Landau levels occupied by electrons for a given density and temperature. For T = 0,

4

νmax =

[

µ2e − m2

e

2eBm

]

,

where [ ] indicates the nearest integer less than the actual value.To obtain an expression for bulk viscosity coefficient, we next consider the pressure tensor, given by

Πµν = ∆µσT στ∆ν

τ + P∆µν (24)

Where the reversible part

Π(0)µν = 0 (25)

and the non-equilibrium part

Π(1)µν = ∆µσT στ(1)∆ν

τ

=eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0∆µ

σ∆ντpσpτχ(x, p)f (0)(p) (26)

Hence omitting 1, we have the traceless part of pressure tensor

Π = −1

3Πµ

µ

= −1

3

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0∆σµ∆µ

τ pσpτf (0)(p)χ(x, p)

= ηv∇µuµ (27)

Hence we have the bulk viscosity coefficient

ηv =1

3T

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p20

∆τσpτpστQ(1 − f (0))f (0) (28)

We shall now calculate the the electrical conductivity for electron gas. The electric four current is given by

jµ(x) =eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0pµf(x, p) (29)

Now because of local charge neutrality, the reversible part jµ(0) = 0 and the non-equilibrium part is given by

jµ(1) = jµ =eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpz

p0pµf (0)χ(x, p) (30)

Then using the covariant form of Ohm’s law

jµ = σµνEν (31)

the electrical conductivity tensor is given by

σµν =e2

T

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

τdpz

p20

pµpν

(

pαuα

h+ 1

)

f (0)(1 − f (0)) (32)

This equation shows that due the presence of strong quantizing magnetic field the electrical conductivity becomes asecond rank tensor- even if the space is isotropic in nature. The zz-component is given by

σzz =e2

T

eBm

4π2

∑

ν

(2 − δν0)

∫ +∞

−∞

dpzτ

p20

p2z

(

1 +pαuα

h

)

f (0)(1 − f (0)) (33)

The ⊥⊥-component is given by

5

σ⊥⊥ =e2

T

eBm

4π2

∑

ν

(2 − δν0)(2νeBm)

∫ +∞

−∞

dpzτ

p20

(

1 +pαuα

h

)

f (0)(1 − f (0)) (34)

and finally the ⊥ z-component is given by

σ⊥z = σz⊥ =e2

T

eBm

4π2

∑

ν

(2 − δν0)(2νeBm)1/2

∫ +∞

−∞

dpzτpz

p20

(

1 +pαuα

h

)

f (0)(1 − f (0)) (35)

From eqn.(34) and (35) it is quite obvious that just like the shear viscosity coefficient, both σ⊥⊥ and σ⊥z componentsof electrical conductivity vanish for νmax = 0, i.e, in the ultra strong magnetic field limit. This is of course not at allevident for ηv, λ and σzz.

3. RATE OF ELECTROMAGNETIC PROCESSES

Now to obtain the numerical values of all these transport coefficients, or their variations with the strength ofmagnetic field we have to know the relaxation time τ , given by

1

τ=

∑

i

Wi (36)

where Wi is the rate of ith electromagnetic process and the sum is over all possible electromagnetic processes takingplace involving the electrons.

Therefore to obtain the relaxation time we evaluate the rates of the basic electromagnetic processes, given bye + e → e + e, and e + p → e + p. Now in the case of e − e-Scattering it is necessary to consider both direct andexchange processes, whereas in the case of e − p scattering only the direct term contributes. Let us first consider thee − e-scattering process, then the direct part is given by

T(d)fi =

ie2

L2yL

2zQ

2(2π)3δ(E1 + E2 − E3 − E4)δ(k1y + k2y − k3y − k4y)δ(k1z + k2z − k3z − k4z)

∫ ∞

−∞

dx (u(k3, x)γµu(k1, x)) (u(k4, x)γµu(k2, x)) (37)

Similarly the exchange term is given by

T(ex)fi =

−ie2

L2yL

2zQ

2(2π)3δ(E1 + E2 − E3 − E4)δ(k1y + k2y − k3y − k4y)δ(k1z + k2z − k3z − k4z)

∫ ∞

−∞

dx (u(k4, x)γµu(k1, x)) (u(k3, x)γµu(k2, x)) (38)

where Q is the exchanged momentum. The rate element is then given by

dW = limt→∞

| T(d)fi + T

(ex)fi |2

t(39)

Since the spinors are functions of x-coordinate through Iν,py(x) (see eqns.(2)-(4)), we use the relation

| f |2=∫ ∞

−∞

dx

∫ ∞

−∞

dx′f∗(x)f(x′) (40)

and we also need the positive energy projection operator, given by

Λ+ =∑

spin

u(k, x)u(k, x′) (41)

Substituting the positive energy spinor solutions (eqns.(2)-(3)), we have

Λ+ =1

2Eν(Akµγµ(µ = 0 and z) + mA + Bkµγµ(µ = y and py = p⊥) (42)

6

The matrices A and B are given by

A =

IνI ′ν 0 0 00 Iν−1I

′ν−1 0 0

0 0 IνI ′ν 00 0 0 Iν−1I

′ν−1

(43)

B =

Iν−1I′ν 0 0 0

0 IνI ′ν−1 0 00 0 Iν−1I

′ν 0

0 0 0 IνI ′ν−1

(44)

where primes indicate the functions of x′. Since the Dirac γ matrices are traceless and both A and B matrices arediagonal in nature with identical blocks we have a few interesting relations, e.g.,

Tr(γµγνA1A2..B1B2..) = Tr(A1A2..B1B2..)gµν , (45)

Tr(γµγνγλγσA1A2..B1B2..) = Tr(A1A2..B1B2..)(gµνgσλ − gµλgνσ + gµλgνσ) (46)

Tr(product of odd no ofγs with any number of A and/or B matrices) = 0 (47)

etc. The beautiful form of A and B matrices allow to multiply γ-matrices in the above relations from any side or inany order. The other interesting aspects of A and B matrices arei) k1µk2µTr(A1A2) = (E1E2 − k1zk2z)Tr(A1A2)

ii) k1µk2µTr(B1B2) = ~k1⊥.~k2⊥Tr(B1B2)iii) k1µk2µTr(A1B2) = k1µk2µTr(B1A2) = 0

iv) p1µk1µp2νk2νTr(A1B2) 6= 0 = (Eν1Eν′

2− p1zk1z)~p2⊥.~k2⊥Tr(A1B2)

All these results are entirely new and to our knowledge not reported before in the literature. We do believe thatthese new results can have interesting applications in various studies of properties of strongly magnetized dense stellarmatter. With all these new formulae, the transition matrix element for e e scattering direct term is given by

Wee;d =∑

ν2

∑

ν3

∑

ν4

∫

dxdx′ e4

(2π)31

4δ(Eν1

+ Eν2− Eν3

− Eν4)δ(p1y + p2y − p3y − p4y)δ(p1z + p2z − p3z − p4z)

γν2γν3

γν4

1

(p1 − p3)4× 1

8Eν1Eν2

Eν3Eν4

[(k3.k4)(k1.k2)

{Tr(A3A1)Tr(A4A2) + Tr(A3B1)Tr(A4B2) + Tr(B3A1)Tr(B4A2) + Tr(B3B1)Tr(B4B2)}+(k3.k2)(k1.k4){Tr(A3A1)Tr(A4A2) + Tr(A3B1)Tr(B4A2) + Tr(B3A1)Tr(A4B2) + Tr(B3B1)Tr(B4B2)}−m2

e(k3.k1){Tr(A3A1)Tr(A4A2) + Tr(B3B1)Tr(A4A2)} − m2e(k4.k2){Tr(A3A1)Tr(B4B2)

+Tr(A3A1)Tr(A4A2)} + 2m4e{Tr(A3A1)Tr(A4A2)}]dp2ydp2zdp3ydp3zdp4ydp4z

f0(p2z)(1 − f0(p3z))(1 − f0(p4z)) (48)

where γνi= (2 − δνi

0).The exchange term is given by

Wee;ex =∑

ν2

∑

ν3

∑

ν4

∫

dxdx′ e4

(2π)31

4δ(Eν1

+ Eν2− Eν3

− Eν4)δ(p1y + p2y − p3y − p4y)δ(p1z + p2z − p3z − p4z)

γν2γν3

γν4

1

(p1 − p4)4× 1

8Eν1Eν2

Eν3Eν4

[(k3.k4)(k1.k2)

{Tr(A1A4)Tr(A2A3) + Tr(B1A4)Tr(B2A3) + Tr(A1B4)Tr(A2B3) + Tr(B1B4)Tr(B2B3)}+ (k1.k3)(k2.k4){Tr(A1A4)Tr(A2A3) + Tr(B1A4)Tr(A2B3) + Tr(A1B4)Tr(B2A3) + Tr(B1B4)Tr(B2B3)}− m2

e(k1.k4){Tr(A1A4)Tr(A2A3) + Tr(B1B4)Tr(A2A3)}− m2

e(k2.k3){Tr(A1A4)Tr(A2A3) + Tr(A1A4)Tr(B2B3)}+ 2m4

eTr(A1A4)Tr(A2A3)]dp2ydp2zdp3ydp3zdp4ydp4z

f0(p2z)(1 − f0(p3z))(1 − f0(p4z)) (49)

7

and similarly the e e-scattering mixed term is given by

Wee;mix =∑

ν2

∑

ν3

∑

ν4

∫

dxdx′ e4

(2π)31

4δ(Eν1

+ Eν2− Eν3

− Eν4)δ(p1y + p2y − p3y − p4y)δ(p1z + p2z − p3z − p4z)

γν2γν3

γν4

1

(p1 − p4)2(p1 − p3)2× 1

8Eν1Eν2

Eν3Eν4

[−4(k3.k4)(k1.k2)

{Tr(A1A2A3A4) + Tr(A1A2B3B4) + Tr(B1B2A3A4) + Tr(B1B2B3B4)}+ 2m2

e(k1.k3){Tr(A1A2A3A4) + Tr(B1A2B3A4) + 2m2e(k1.k2){Tr(A1A2A3A4) + Tr(B1B2A3A4)}]

+ 2m2e(k1.k4){Tr(A1A2A3A4) + Tr(B1A2A3B4)} + 2m2

e(k2.k3){Tr(A1A2A3A4) + Tr(A1B2B3B4)}+ 2m2

e(k3.k4){Tr(A1A2A3A4) + Tr(A1A2B3B4)} +

2m2e(k2.k4){Tr(A1A2A3A4)Tr(A1B2A3B4)} − m4

eTr(A1A2A3A4)]

dp2ydp2zdp3ydp3zdp4ydp4zf0(p2z)(1 − f0(p3z))(1 − f0(p4z)) (50)

and finally the rate of e p-scattering is given by

Wep =∑

ν2

∑

ν3

∑

ν4

∫

dxdx′ e4

(2π)31

4δ(E1 + E2 − E3 − E4)δ(p1y + p2y − p3y − p4y)δ(p1z + p2z − p3z − p4z)

γν2γν3

γν4

1

(p1 − p3)4× 1

8Eν1Eν2

Eν3Eν4

[(k3.k4)(k1.k2)

{Tr(A3A1)Tr(A4A2) + Tr(A3B1)Tr(A4B2) + Tr(B3A1)Tr(B4A2) + Tr(B3B1)Tr(B4B2)}+(k3.k2)(k1.k4){Tr(A3A1)Tr(A4A2) + Tr(A3B1)Tr(B4A2) + Tr(B3A1)Tr(A4B2)

+Tr(B3B1)Tr(B4B2)} − m2p(k3.k1){Tr(A3A1)Tr(A4A2) + Tr(B3B1)Tr(A4A2)}

−m2e(k4.k2){Tr(A3A1)Tr(B4B2) + Tr(A3A1)Tr(A4A2)} + 2m2

em2p{Tr(A3A1)Tr(A4A2)}]

dp2ydp2zdp3ydp3zdp4ydp4zf0(p2z)(1 − f0(p3z))(1 − f0(p4z)) (51)

To evaluate the multidimensional (eight dimensional) integrals, we have used three δ-functions and finally evaluatenumerically using multi-dimensional Monte-Carlo integration technique and obtain the rates from the eqns.(48)-(51).We have generated the Hermite polynomials appearing in the matrices A and B numerically. We obtain the relaxationtime from eqn.(36) and finally evaluate the kinetic coefficients from eqns.(18),(23),(28),(33)-(35) for various valuesof magnetic fields Bm, temperature T and matter density nB. In figs.(1)-(6) we have shown the variation of kineticcoefficients with magnetic field strength. Identical results can also be obtained for dense proton matter.

4. DISCUSSIONS

From the figs.(1)-(6) it is quite obvious that the kinetic coefficients are almost independent of magnetic field formoderate strengths (< 1017G), but all of them go to almost zero for magnetic field strength beyond 1017G. Thefirst conclusion is therefore, that at ultra strong magnetic field, the matter (electron or proton matter) behaves likea superfluid insulator. The mechanism of superfluidity is of course completely different from conventional neutronmatter.

Now as we know from classical plasma physics in presence of strong magnetic field, that the charged particles canonly travel along the lines of force, motions are almost forbidden across the field, as a consequence σ⊥⊥ or σ⊥z vanishfor a classical plasma in presence of ultra-strong magnetic field, whereas σzz remains non-zero even at very highmagnetic field strength. On the other hand, in the case of quantum mechanical plasma as the magnetic field becomestrong enough, the maximum value of Landau quantum number νmax → 0, which further means the system becomeseffectively one dimensional in the ultra-strong magnetic field limit. Then in the collision dominated scenario, since pz

varies continuously from −∞ to +∞ and p⊥ → 0, σzz → 0. To elaborate this point a bit- since the current is flowingalong the field lines only, then from the symmetric nature of pz, we have

j(+)z = −j(−)

z =⇒ jz = j(+)z + j(−)

z = 0 (52)

This is specially true for an effectively one dimensional collision dominated quantum plasma. In the case one dimen-sional flow (free streaming) of charged particles along the field lines, this is not true; σzz has non-zero finite value.Since all the three component of electrical conductivity vanish in presence of ultra-relativistic magnetic field, this

8

must affect significantly the Ohmic decay of strong magnetic field. We know that the Ohmic decay time scale is givenby

τd =4πσl2

c2(53)

where l is the dimension of the system. Since σ → 0 for extremely strong magnetic field, as a consequence, thefield should decay quickly. We have noticed that a field of strength 1018G become ∼ 1015G within a minute. Thismoderate field of course will take longer time to decay further (long decay time scale). Therefore the last conclusionis that stronger the magnetic field, shorter the decay time scale and as a result we do believe that the magnetic fieldstrength at the core region of a magnetar can not be high enough and consequently there will be no problem on themechanical stability or the existence of magnetars.

FIG. 1. Variation of shear viscosity coefficient with magnetic field B. Curve (a): T = 15MeV and nB = 3n0, Curve (b):T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

9

FIG. 2. Variation of bulk viscosity coefficient with magnetic field B. Curve (a): T = 15MeV and nB = 3n0, Curve (b):T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

FIG. 3. Variation of heat conductivity coefficient with magnetic field B. Curve (a): T = 15MeV and nB = 3n0, Curve (b):T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

10

FIG. 4. Variation of zz component of electrical conductivity with magnetic field B. Curve (a): T = 15MeV and nB = 3n0,Curve (b): T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

FIG. 5. Variation of pp component of electrical conductivity with magnetic field B. Curve (a): T = 15MeV and nB = 3n0,Curve (b): T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

11

FIG. 6. Variation of zp component of electrical conductivity with magnetic field B. Curve (a): T = 15MeV and nB = 3n0,Curve (b): T = 15MeV and nB = 5n0, and Curve (c): T = 30MeV and nB = 3n0,

[1] R.C. Duncan and C. Thompson, Astrophys. J. Lett. 392, L9 (1992); C. Thompson and R.C. Duncan, Astrophys. J. 408,194 (1993); C. Thompson and R.C. Duncan, MNRAS 275, 255 (1995); C. Thompson and R.C. Duncan, Astrophys. J.473, 322 (1996).

[2] P.M. Woods et al., Astrophys. J. Lett. 519, L139 (1999); C. Kouveliotou, et al., Nature 391, 235 (1999).[3] K. Hurley, et al., Astrophys. Jour. 442, L111 (1999).[4] S. Mereghetti and L. Stella, Astrophys. Jour. 442, L17 (1999); J. van Paradihs, R.E. Taam and E.P.J. van den Heuvel,

Astron. Astrophys. 299, L41 (1995); S. Mereghetti, astro-ph/99111252; see also A. Reisenegger, astro-ph/01003010.[5] D. Bandopadhyaya, S. Chakrabarty, P. Dey and S. Pal, Phys. Rev. D58, 121301 (1998).[6] S. Chakrabarty, D. Bandopadhyay and S. Pal, Phys. Rev. Lett. 78, 2898 (1997); D. Bandopadhyay, S. Chakrabarty and

S. Pal, Phys. Rev. Lett. 79, 2176 (1997).[7] C.Y. Cardall, M. Prakash and J.M. Lattimer, astro-ph/0011148 and references therein; E. Roulet, astro-ph/9711206; L.B.

Leinson and A. Perez, astro-ph/9711216; D.G. Yakovlev and A.D. Kaminkar, The Equation of States in Astrophysics, eds.G. Chabrier and E. Schatzman P.214, Cambridge Univ.

[8] S. Chakrabarty and P.K. Sahu, Phys. Rev. D53, 4687 (1996).[9] S. Chakrabarty, Phys. Rev. D51, 4591 (1995); Chakrabarty, Phys. Rev. D54, 1306 (1996).

[10] T. Ghosh and S. Chakrabarty, Phys. Rev. D63, 0403006 (2001); T. Ghosh and S. Chakrabarty, Int. Jour. Mod. Phys.D10, 89 (2001).

[11] V.G. Bezchastrov and P. haensel, astro-ph/9608090.[12] S. Mandal and S. Chakrabarty (two papers on weak and electromagnetic processes to be published).[13] A. Melatos, Astrophys. Jour. 519, L77 (1999); A. Melatos, MNRAS 313, 217 (2000).[14] S. Bonazzola et al, Astron. & Atrophysics. 278, 421 (1993); M. Bocquet et al, Astron. & Atrophysics. 301, 757 (1995).[15] B. Bertotti and A.M. Anile, Astron. & Atrophysics. 28, 429 (1973); C. Cutler and D.I. Jones, gr-qc/0008021; K. Konno,

T. Obata and Y. Kojima, gr-qc/9910038; A.P. Martinez et al, hep-ph/0011399; M. Chaichian et al, Phys. Rev. Lett. 84,5261 (2000); Guangjun Mao, Akira Iwamoto and Zhuxia Li, astro-ph/0109221; A. Melatos, Astrophys. Jour. 519, L77(1999); A. Melatos, MNRAS 313, 217 (2000); R. Gonzalez Felipe et al, astro-ph/0207150 and references therein.

[16] S. Mandal and S. Chakrabarty (submitted).[17] S. Ghosh, S. Mandal and S. Chakrabarty (Ann. Phys. in press).[18] S.R. de Groot, W.A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory, North Holland, Amsterdam, 1980.

12